Long-range higher-order surface-plasmon polaritons

Besides the two fundamental symmetric and antisymmetric modes, a thin metal slab surrounded by a dielectric medium supports an infinite number of other modes whose propagation distances, however, are extremely short. We show that such a higher-order mode, normally not regarded to be useful, can turn into a long-range surface-plasmon polariton in circumstances where the fundamental long-range mode does not exist and the propagation length of the single-interface surface-plasmon polariton is negligible. This transition, which is an electromagnetic near-field phenomenon not previously observed, occurs with a broad range of bandwidths and materials, especially in the context of high-index semiconductors. The long-range behavior of the higher-order surface-plasmon polaritons does not originate from the coupling between the polaritons supported by the individual boundaries, and in some situations it comes with increased field confinement, which is the opposite of what occurs for the fundamental long-range surface mode. Our results, which follow from rigorous electromagnetic theory, provide deeper insight into the foundations of thin-film plasmonics and suggest avenues for enhanced properties of nanophotonic devices involving modern polaritonic media.

Figure 1

Illustration of SPPs at a metal film of thickness $d$ surrounded by a nonabsorbing dielectric possessing relative permittivities ${\epsilon }_{r1}$ (complex) and ${\epsilon }_{r2}$ (real). With time dependence $e^{-i\omega t}$, the spatial parts of the $p$-polarized electric fields are ${\mathbf{E}}_2^{\left(1\right)}\left(\mathbf{r}\right)=E_2^{\left(1\right)}{e}^{i{\mathbf{k}}_2^{\left(1\right)}·\mathbf{r}}{\widehat{\mathbf{p}}}_2^{\left(1\right)} \left(z\ge d/2\right),{\mathbf{E}}_1\left(\mathbf{r}\right)=E_1^{\left(1\right)}{e}^{i{\mathbf{k}}_1^{\left(1\right)}·\mathbf{r}}{\widehat{\mathbf{p}}}_1^{\left(1\right)}+E_1^{\left(2\right)}{e}^{i{\mathbf{k}}_1^{\left(2\right)}·\mathbf{r}}{\widehat{\mathbf{p}}}_1^{\left(2\right)} \left(\right|z|<d/2)$, and ${\mathbf{E}}_2^{\left(2\right)}\left(\mathbf{r}\right)=E_2^{\left(2\right)}{e}^{i{\mathbf{k}}_2^{\left(2\right)}·\mathbf{r}}{\widehat{\mathbf{p}}}_2^{\left(2\right)} \left(z\le d/2\right)$, where $E_{\alpha }^{\left(\beta \right)}$ is a complex amplitude with $\alpha ,\beta \in \{1,2\}$. The wave vector ${\mathbf{k}}_{\alpha }^{\left(\beta \right)}$, for which ${\mathbf{k}}_{\alpha }^{\left(\beta \right)}·{\mathbf{k}}_{\alpha }^{\left(\beta \right)}=k_{\alpha }^2=k_0^2{\epsilon }_{r\alpha }$ ($k_{\alpha }$ and $k_0$ are the wave numbers in medium $\alpha$ and vacuum, respectively), and the nor-malized polarization vector ${\widehat{\mathbf{p}}}_{\alpha }^{\left(\beta \right)}={\widehat{\mathbf{k}}}_{\alpha }^{\left(\beta \right)}\times {\widehat{\mathbf{e}}}_y$, with ${\widehat{\mathbf{e}}}_y$ being the unit vector in the positive $y$ direction and ${\widehat{\mathbf{k}}}_{\alpha }^{\left(\beta \right)}={\mathbf{k}}_{\alpha }^{\left(\beta \right)}{[{\mathbf{k}}_{\alpha }^{\left(\beta \right)*}·{\mathbf{k}}_{\alpha }^{\left(\beta \right)}]}^{-1/2}$ (the asterisk denotes complex conjugation), lie in the $xz$ plane. Note that ${\widehat{\mathbf{p}}}_{\alpha }^{\left(\beta \right)}$ and ${\widehat{\mathbf{k}}}_{\alpha }^{\left(\beta \right)}$ are not orthogonal if ${\mathbf{k}}_{\alpha }^{\left(\beta \right)}$ is complex, in which case $|{\mathbf{k}}_{\alpha }^{\left(\beta \right)}|\ne k_{\alpha }$ [17, 18, 24].

Figure 2

(left) Real and (right) imaginary parts of the tangential wave-vector component $k_x$ for the FM (solid line) and the three HOMs with the lowest $k_x^{\prime \prime }$ (dashed, dash-dotted, and dotted lines) of a Ag slab in vacuum as a function of the film thickness $d$ at the vacuum wavelength ${\lambda }_0=2\pi /k_0=500$ nm. The inset on the right demonstrates that $k_x^{\prime \prime }$ for the FM approaches zero when $d\to 0$. The relative permittivity of Ag is ${\epsilon }_{r1}=-8.50+i0.76$ [28].

Figure 3

(left) Real and (right) imaginary parts of the tangential wave-vector component $k_x$ for the FM (solid lines) and the HOM with the lowest $k_x^{\prime \prime }$ (dotted lines) for a Au slab surrounded by GaP as a function of the film thickness $d$ at different vacuum wavelengths ${\lambda }_0=2\pi /k_0$. The relative permittivities of Au and GaP are from [28] and [29], respectively.

Figure 4

Dispersion relations for the FM (solid curves) and HOM (dotted curves) corresponding to Fig. 3 at different slab thicknesses. The middle row shows a close-up view of the mode flip in the vicinity of the transition point (the horizontal dotted lines mark the transition frequency).

Figure 5

Longitudinal propagation length $l_x$ for the FMs (solid curves) and HOMs (dotted curves) corresponding to Fig. 3 as a function of the free-space wavelength ${\lambda }_0$ for selected slab thicknesses: 10 nm (dark lines), 15 nm (medium lines), and 20 nm (light lines). The dashed curve in the left panel is an approximate expression that is valid in the limit of thin films [25], evaluated for a thickness of 10 nm, and the dash-dotted line in the right panel shows the corresponding single-interface SPP. Observe the different scale for $l_x$ in the right panel, which gives an expanded view of the modes with a nearly zero propagation distance. The vertical dotted lines show the transition wavelength.

Figure 6

Longitudinal propagation length $l_x$ for the FM (solid curves) and the HOM of the lowest damping (dotted curves) for a Ag slab surrounded by ${\text{SiO}}_2$ as a function of the free-space wavelength ${\lambda }_0$ for selected film thicknesses: 10 nm (dark lines), 15 nm (medium lines), and 20 nm (light lines). The dashed line in the left panel is the approximate thin-film expression [25] for a thickness of 10 nm, the dash-dotted curve in the right panel corresponds to the single-interface SPP, and the vertical dotted lines show the transition wavelengths. Notice the different scales for $l_x$ in the left and right panels (see Fig. 5). The relative permittivities for Ag and ${\text{SiO}}_2$ are from [28].

Figure 7

Dispersion relations for the FMs (solid curves) and HOMs (dotted curves) of (top) Fig. 5 and (bottom) Fig. 6. Observe the different $k_x^{\prime }/k_0$ scales in the right column, showing extended views of the dispersion curves near the light lines of the surroundings (dashed lines). The horizontal dotted lines mark the transition frequencies.

Figure 8

Dispersion relations of the long-range HOMs for (left) Au/GaP and (right) $\mathrm{Ag}/{\text{SiO}}_2$ at different film thicknesses: 10 nm (dark lines), 15 nm (medium dark lines), 20 nm (medium light lines), and 25 nm (light lines). The dashed lines are the light lines of the surroundings.

Figure 9

(top) SPP wavelength ${\lambda }_{\mathrm{SPP}}$ and penetration depths (bottom left) $l_{1z}$ inside and (bottom right) $l_{2z}$ outside the slab for the FMs (solid curves) and HOMs (dotted curves) corresponding to Fig. 5 as a function of the slab thickness $d$ for selected vacuum wavelengths: 580 nm (dark lines), 610 nm (medium lines), and 640 nm (light lines). In all cases the HOM represents the long-range SPP guidance, whereas the fundamental LRSPP does not exist. The dashed lines corresponding to the respective single-interface SPPs are included for reference.

Figure 10

(left) Ratio of the strengths $|E_{2x}^{\left(1\right)}|/|E_{2z}^{\left(1\right)}|$ and (right) the phase difference $\Delta {\varphi }_2^{\left(1\right)}$ between the $x$ and $z$ components of the electric field above the slab for the FMs (solid curves) and HOMs (dotted curves) in (top) Fig. 5 and (bottom) Fig. 6 as a function of the free-space wavelength ${\lambda }_0$ for selected film thicknesses: 10 nm (dark lines), 15 nm (medium lines), and 20 nm (light lines). The vertical dotted lines show the transition wavelengths, and the dashed curves correspond to the respective single-interface SPPs.

Figure 11

Examples of some materials and bandwidths for which the lowest-order HOM amounts to the LRSPP. The relative permittivities for ${\text{SiO}}_2$, Ag, Au, Cu, and Na are from [28], and those for GaP, ZnO, and Al are from [29], [32], and [33], respectively.